Now that we know how to represent our state vector as a superposition
of states, and we know that we can only measure the state vector to be
in one of the base states, it would seem that there would be some sort
of discrepancy. We must determine what happens when we measure the
state vector. We know from quantum physics that given an initial
condition the state vector will evolve in time in accordance with
Schrödenger's equation:

ih = H(t)| X(t)

Where i is the square root of negative one, h is
1.0545*10-34
Js, and H is the Hamiltonian operator, which is determined by the
physical characteristics of the system being evolved.

In our notation this expression is:

ih = H(t)ij*wj(t)

This evolution would appear to be continuous, but these equations only
apply to a quantum mechanical system evolving in isolation from the
environment. The only way to observe the state of the state vector is
to in some way cause the quantum mechanical system to interact with
the environment. When the state vector is observed it makes a sudden
discontinuous jump to one of the eigenstates, but this is not a
violation of Schrödenger's equation. When the quantum mechanical
system interacts with the outside environment and measured the state
vector is said to have collapsed. (Williams, Clearwater)

Now understanding that a state vector will collapse when it interacts
with the external environment, we still need to know in what manner
this collapse happens. To perform any sort of useful calculation we
must be able so say something about which base state a quantum
mechanical system will collapse into. The probability that the state
vector will collapse into the j'th eigenstate, is given by
| wj|2 which is defined to be
aj2 + bj2 if
wj = aj + i*bj, where wj is the complex projection of the
state vector onto the j'th eigenstate. In general the chance of
choosing any given state is

Prob(j) =

but
as mentioned earlier we will insist on having the state vector of
length one, and in this case the probability expression simplifies to
Prob(j) = | wj|2.

So now we know how to construct an n state quantum system, which can
be placed in an arbitrary superposition of states. We also know how
to measure the resultant superposition and get a base state with a
certain probability. This is all that we need to understand about our
quantum memory register to be able to simulate its behavior.